When can one use a non-stationary VAR validly? In Chris Brooks book Introductory Econometrics for Finance on page 292 when discussing the problem with VARS he mentions.

... many proponents of the VAR approach recommend that differencing to induce stationarity should not be done. They would argue that the purpose of VAR estimation is purely to examine the relationship between the variables, and that differencing will throw away information on any long-run relationship between the series away..  

Who are these "proponents of the VAR approach" and under what circumstances can I use a Non-stationary VAR? 
 A: I am guessing that the thrown away information the author is referring to is alluding to the situation where the VAR model is cointegrated. This is one situation, but there might be others. 
Start with a non-stationary VAR(2)
$$
Y_t = A_1 Y_{t-1} + A_2 Y_{t-2} + U_t.
$$
Then difference once to get
\begin{align*}
\Delta Y_t &= -IY_{t-1} + A_1 Y_{t-1} + A_2 Y_{t-2} + U_t \\
 &= -IY_{t-1} + A_1 Y_{t-1} + A_2 Y_{t-1} - A_2 Y_{t-1} + A_2 Y_{t-2} + U_t\\
 &= -(I - A_1 - A_2)Y_{t-1} - A_2 \Delta Y_{t-1} + W_t \\
 &= \alpha \beta ' Y_{t-1} + \Gamma_1 \Delta Y_{t-1} + U_t.
\end{align*}
$(I - A_1 - A_2) = \alpha \beta '$ is singular because the determinant AR polynomial has a unit root. If you differenced $Y_t$ and estimated a VAR(1), you would be ignoring/omitting the $\alpha \beta ' Y_{t-1}$ term which describes an economic equilibrium between some of the elements of $Y_t$. 
I'm looking at the table of contents for this book online right now, and this sort of stuff is discussed a bit later on. Keep reading; it looks like it will be in chapter 8, if you have the same edition.
